MathDB

1

Polish MO Finals 2018, Problem 6

p>3

is given. Let

K

be the number of such permutations

(a_1, a_2, \ldots, a_p)

of

\{ 1, 2, \ldots, p\}

such that

a_1a_2+a_2a_3+\ldots + a_{p-1}a_p+a_pa_1

is divisible by

p

. Prove

K+p

is divisible by

p^2

.

5

1

Polish MO Finals 2018, Problem 5

An acute triangle

ABC

in which

AB<AC

is given. Points

E

and

F

are feet of its heights from

B

and

C

, respectively. The line tangent in point

A

to the circle escribed on

ABC

crosses

BC

at

P

. The line parallel to

BC

that goes through point

A

crosses

EF

at

Q

. Prove

PQ

is perpendicular to the median from

A

of triangle

ABC

.

4

1

Polish MO Finals 2018, Problem 4

Let

n

be a positive integer. Suppose there are exactly

M

squarefree integers

k

such that

\left\lfloor\frac nk\right\rfloor

is odd in the set

\{ 1, 2,\ldots, n\}

. Prove

M

is odd.
An integer is squarefree if it is not divisible by any square other than

1

.

3

1

Polish MO Finals 2018, Problem 3

Find all real numbers

c

for which there exists a function

f\colon\mathbb R\rightarrow \mathbb R

such that for each

x, y\in\mathbb R

it's true that

f(f(x)+f(y))+cxy=f(x+y).

2

1

Polish MO Finals 2018, Problem 2

A subset

S

of size

n

of a plane consisting of points with both coordinates integer is given, where

n

is an odd number. The injective function

f\colon S\rightarrow S

satisfies the following: for each pair of points

A, B\in S

, the distance between points

f(A)

and

f(B)

is not smaller than the distance between points

A

and

B

. Prove there exists a point

X

such that

f(X)=X

.

1

Polish MO Finals 2018, Problem 1

An acute triangle

ABC

in which

AB<AC

is given. The bisector of

\angle BAC

crosses

BC

at

D

. Point

M

is the midpoint of

BC

. Prove that the line though centers of circles escribed on triangles

ABC

and

ADM

is parallel to

AD

.

2018 Polish MO Finals

Subcontests

Polish MO Finals 2018, Problem 6

Polish MO Finals 2018, Problem 5

Polish MO Finals 2018, Problem 4

Polish MO Finals 2018, Problem 3

Polish MO Finals 2018, Problem 2

Polish MO Finals 2018, Problem 1